A structural approach to matching problems with preferences

This thesis is a study of a number of matching problems that seek to match together pairs or groups of agents subject to the preferences of some or all of the agents. We present a number of new algorithmic results for five specific problem domains. Each of these results is derived with the aid of some structural properties implicitly embedded in the problem. We begin by describing an approximation algorithm for the problem of finding a maximum stable matching for an instance of the stable marriage problem with ties and incomplete lists (MAX-SMTI). Our polynomial time approximation algorithm provides a performance guarantee of 3/2 for the general version of MAX-SMTI, improving upon the previous best approximation algorithm, which gave a performance guarantee of 5/3. Next, we study the sex-equal stable marriage problem (SESM). We show that SESM is W[1]-hard, even if the men's and women's preference lists are both of length at most three. This improves upon the previously known hardness results. We contrast this with an exact, low-order exponential time algorithm. This is the first non-trivial exponential time algorithm known for this problem, or indeed for any hard stable matching problem. Turning our attention to the hospitals / residents problem with couples (HRC), we show that HRC is NP-complete, even if very severe restrictions are placed on the input. By contrast, we give a linear-time algorithm to find a stable matching with couples (or report that none exists) when stability is defined in terms of the classical Gale-Shapley concept. This result represents the most general polynomial time solvable restriction of HRC that we are aware of. We then explore the three dimensional stable matching problem (3DSM), in which we seek to find stable matchings across three sets of agents, rather than two (as in the classical case). We show that under two natural definitions of stability, finding a stable matching for a 3DSM instance is NP-complete. These hardness results resolve some open questions in the literature. Finally, we study the popular matching problem (POP-M) in the context of matching a set of applicants to a set of posts. We provide a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a new structure called the switching graph exploited to yield efficient algorithms for a range of associated problems, extending and improving upon the previously best-known results for this problem

The Next Generation Virgo Cluster Survey. XII. Stellar Populations and Kinematics of Compact, Low-Mass Early-Type Galaxies from Gemini GMOS-IFU Spectroscopy

We present Gemini GMOS-IFU data of eight compact low-mass early-type galaxies (ETGs) in the Virgo cluster. We analyse their stellar kinematics, stellar population, and present two-dimensional maps of these properties covering the central 5"x 7" region. We find a large variety of kinematics: from non- to highly-rotating objects, often associated with underlying disky isophotes revealed by deep images from the Next Generation Virgo Cluster Survey. In half of our objects, we find a centrally-concentrated younger and more metal-rich stellar population. We analyze the specific stellar angular momentum through the lambdaR parameter and find six fast-rotators and two slow-rotators, one having a thin counter-rotating disk. We compare the local galaxy density and stellar populations of our objects with those of 39 more extended low-mass Virgo ETGs from the SMAKCED survey and 260 massive ($M>10^{10}$\Msun) ETGs from the A3D sample. The compact low-mass ETGs in our sample are located in high density regions, often close to a massive galaxy and have, on average, older and more metal-rich stellar populations than less compact low-mass galaxies. We find that the stellar population parameters follow lines of constant velocity dispersion in the mass-size plane, smoothly extending the comparable trends found for massive ETGs. Our study supports a scenario where low-mass compact ETGs have experienced long-lived interactions with their environment, including ram-pressure stripping and gravitational tidal forces, that may be responsible for their compact nature.Comment: Accepted in ApJ, 19 pages, 10 figure

Dynamical modelling of stars and gas in NGC2974: determination of mass-to-light ratio, inclination and orbital structure by Schwarzschild's method

We study the large-scale stellar and gaseous kinematics of the E4 galaxy NGC2974, based on panoramic integral-field data obtained with SAURON. We quantify the velocity fields with Fourier methods (kinemetry), and show that the large-scale kinematics is largely consistent with axisymmetry. We construct general axisymmetric dynamical models for the stellar motions using Schwarzschild's orbit-superposition method, and compare the inferred inclination and mass-to-light ratio with the values obtained by modelling the gas kinematics. Both approaches give consistent results. However we find that the stellar models provide fairly weak constraints on the inclination. The intrinsic orbital distribution of NGC2974, which we infer from our model, is characterised by a large-scale stellar component of high angular momentum. We create semi-analytic test models, resembling NGC2974, to study the ability of Schwarzschild's modelling technique to recover the given input parameters (mass-to-light ratio and inclination) and the distribution function. We also test the influence of a limited spatial coverage on the recovery of the distribution function (i.e. the orbital structure). We find that the models can accurately recover the input mass-to-light ratio, but we confirm that even with perfect input kinematics the inclination is only marginally constrained. This suggests a possible degeneracy in the determination of the inclination, but further investigations are needed to clarify this issue. For a given potential, we find that the analytic distribution function of our test model is well recovered by the three-integral model within the spatial region constrained by integral-field kinematics.Comment: 22 pages, 24 figures. Accepted for publication in MNRAS. Version with full resolution images available at http://www.strw.leidenuniv.nl/sauron/papers/krajnovic2004_ngc2974.pd

Probing the stellar populations of early-type galaxies: the SAURON survey

The SAURON project will deliver two-dimensional spectroscopic data of a sample of nearby early-type galaxies with unprecedented quality. In this paper, we focus on the mapping of their stellar populations using the SAURON data, and present some preliminary results on a few prototypical cases.Comment: 12 pages, 6 figures. ASP Conference, Galaxies: the Third Dimension, Cozumel. Version with higher resolution figures available at http://www-obs.univ-lyon1.fr/eric.emsellem/papers/cozumel_emsellem.ps.g

Morphology and kinematics of the ionised gas in early-type galaxies

We present results of our ongoing study of the morphology and kinematics of the ionised gas in 48 representative nearby elliptical and lenticular galaxies using the SAURON integral-field spectrograph on the 4.2m William Herschel Telescope. Making use of a recently developed technique, emission is detected in 75% of the galaxies. The ionised-gas distributions display varied morphologies, ranging from regular gas disks to filamentary structures. Additionally, the emission-line kinematic maps show, in general, regular motions with smooth variations in kinematic position angle. In most of the galaxies, the ionised-gas kinematics is decoupled from the stellar counterpart, but only some of them present signatures of recent accretion of gaseous material. The presence of dust is very common in our sample and is usually accompanied by gas emission. Our analysis of the [OIII]/Hbeta emission-line ratios, both across the whole sample as well as within the individual galaxies, suggests that there is no unique mechanism triggering the ionisation of the gas.Comment: 8 pages, 2 figures, submitted to "Adaptive Optics-Assisted Integral-Field Spectroscopy", Rutten R.G.M., Benn C.R., Mendez J., eds., May 2005, La Palma (Spain), New Astr. Rev. For full resolution PS, see http://www.strw.leidenuniv.nl/~jfalcon/JFB_AOmeeting_color_hires.ps.g

Supermassive black holes from OASIS and SAURON integral-field kinematics

Supermassive black holes are a key element in our understanding of how galaxies form. Most of the progress in this very active field of research is based on just ~30 determinations of black hole mass, accumulated over the past decade. We illustrate how integral-field spectroscopy, and in particular our OASIS modeling effort, can help improve the current situation.Comment: 4 pages, 2 figures, LaTeX. To appear in the proceedings of IAU Symposium 245 "Formation and Evolution of Galaxy Bulges", M. Bureau, E. Athanassoula, and B. Barbuy, ed

Fast and Slow Rotators: The build-up of the Red Sequence

Using the unique dataset obtained within the course of the SAURON project, a radically new view of the structure, dynamics and stellar populations of early-type galaxies has emerged. We show that galaxies come in two broad flavours (slow and fast rotators), depending on whether or not they exhibit clear large-scale rotation, as indicated via a robust measure of the specific angular momentum of baryons. This property is also linked with other physical characteristics of early-type galaxies, such as: the presence of dynamically decoupled cores, orbital structure and anisotropy, stellar populations and dark matter content. I here report on the observed link between this baryonic angular momentum and a mass sequence, and how this uniquely relates to the building of the red sequence via dissipative/dissipationless mergers and secular evolution.Comment: 4 pages, 1 figure. To appear in the Proceedings of IAU Symposium 245 "Formation and Evolution of Galaxy Bulges", Eds M. Bureau, E. Athanassoula, and B. Barbu